# Properties

 Label 8281.2.a.cd.1.3 Level $8281$ Weight $2$ Character 8281.1 Self dual yes Analytic conductor $66.124$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$8281 = 7^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8281.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$66.1241179138$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.4507648.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 637) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.90903$$ of defining polynomial Character $$\chi$$ $$=$$ 8281.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.264627 q^{2} +2.90903 q^{3} -1.92997 q^{4} +1.43515 q^{5} -0.769807 q^{6} +1.03998 q^{8} +5.46247 q^{9} +O(q^{10})$$ $$q-0.264627 q^{2} +2.90903 q^{3} -1.92997 q^{4} +1.43515 q^{5} -0.769807 q^{6} +1.03998 q^{8} +5.46247 q^{9} -0.379780 q^{10} -5.50474 q^{11} -5.61435 q^{12} +4.17491 q^{15} +3.58474 q^{16} +4.83072 q^{17} -1.44552 q^{18} -2.82036 q^{19} -2.76981 q^{20} +1.45670 q^{22} -5.99956 q^{23} +3.02532 q^{24} -2.94033 q^{25} +7.16341 q^{27} +1.04188 q^{29} -1.10479 q^{30} -9.20895 q^{31} -3.02857 q^{32} -16.0135 q^{33} -1.27834 q^{34} -10.5424 q^{36} -0.612497 q^{37} +0.746342 q^{38} +1.49252 q^{40} +10.6196 q^{41} -8.43685 q^{43} +10.6240 q^{44} +7.83949 q^{45} +1.58764 q^{46} -2.40922 q^{47} +10.4281 q^{48} +0.778091 q^{50} +14.0527 q^{51} -1.82959 q^{53} -1.89563 q^{54} -7.90015 q^{55} -8.20452 q^{57} -0.275709 q^{58} +0.870914 q^{59} -8.05746 q^{60} -3.33253 q^{61} +2.43693 q^{62} -6.36804 q^{64} +4.23759 q^{66} +6.62741 q^{67} -9.32316 q^{68} -17.4529 q^{69} +6.85856 q^{71} +5.68083 q^{72} -3.14147 q^{73} +0.162083 q^{74} -8.55353 q^{75} +5.44322 q^{76} -17.5723 q^{79} +5.14465 q^{80} +4.45118 q^{81} -2.81022 q^{82} -11.4525 q^{83} +6.93283 q^{85} +2.23261 q^{86} +3.03086 q^{87} -5.72479 q^{88} +0.995318 q^{89} -2.07454 q^{90} +11.5790 q^{92} -26.7891 q^{93} +0.637545 q^{94} -4.04765 q^{95} -8.81020 q^{96} -13.5090 q^{97} -30.0695 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9}+O(q^{10})$$ 6 * q + 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 $$6 q + 8 q^{3} + 4 q^{4} - 6 q^{5} - 4 q^{6} + 6 q^{9} + 4 q^{10} - 4 q^{11} - 4 q^{12} - 12 q^{15} + 16 q^{17} + 4 q^{18} - 2 q^{19} - 16 q^{20} - 12 q^{22} - 6 q^{23} - 12 q^{24} - 4 q^{25} + 20 q^{27} - 6 q^{29} - 6 q^{31} + 20 q^{32} - 4 q^{33} - 24 q^{36} + 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} - 14 q^{45} - 8 q^{46} - 30 q^{47} - 8 q^{48} - 8 q^{50} - 4 q^{51} - 14 q^{53} + 48 q^{54} - 8 q^{55} - 4 q^{57} + 8 q^{58} - 24 q^{59} - 12 q^{60} + 28 q^{62} - 20 q^{64} - 4 q^{66} - 16 q^{67} + 28 q^{68} - 20 q^{69} - 8 q^{71} - 28 q^{72} + 6 q^{73} - 12 q^{74} + 12 q^{75} + 16 q^{76} - 22 q^{79} + 28 q^{80} + 46 q^{81} - 40 q^{82} - 50 q^{83} + 8 q^{85} + 16 q^{86} - 16 q^{87} - 44 q^{88} - 26 q^{89} - 40 q^{90} + 20 q^{92} - 16 q^{93} - 32 q^{94} - 6 q^{95} + 20 q^{96} + 14 q^{97} - 12 q^{99}+O(q^{100})$$ 6 * q + 8 * q^3 + 4 * q^4 - 6 * q^5 - 4 * q^6 + 6 * q^9 + 4 * q^10 - 4 * q^11 - 4 * q^12 - 12 * q^15 + 16 * q^17 + 4 * q^18 - 2 * q^19 - 16 * q^20 - 12 * q^22 - 6 * q^23 - 12 * q^24 - 4 * q^25 + 20 * q^27 - 6 * q^29 - 6 * q^31 + 20 * q^32 - 4 * q^33 - 24 * q^36 + 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 - 14 * q^45 - 8 * q^46 - 30 * q^47 - 8 * q^48 - 8 * q^50 - 4 * q^51 - 14 * q^53 + 48 * q^54 - 8 * q^55 - 4 * q^57 + 8 * q^58 - 24 * q^59 - 12 * q^60 + 28 * q^62 - 20 * q^64 - 4 * q^66 - 16 * q^67 + 28 * q^68 - 20 * q^69 - 8 * q^71 - 28 * q^72 + 6 * q^73 - 12 * q^74 + 12 * q^75 + 16 * q^76 - 22 * q^79 + 28 * q^80 + 46 * q^81 - 40 * q^82 - 50 * q^83 + 8 * q^85 + 16 * q^86 - 16 * q^87 - 44 * q^88 - 26 * q^89 - 40 * q^90 + 20 * q^92 - 16 * q^93 - 32 * q^94 - 6 * q^95 + 20 * q^96 + 14 * q^97 - 12 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.264627 −0.187119 −0.0935596 0.995614i $$-0.529825\pi$$
−0.0935596 + 0.995614i $$0.529825\pi$$
$$3$$ 2.90903 1.67953 0.839765 0.542949i $$-0.182692\pi$$
0.839765 + 0.542949i $$0.182692\pi$$
$$4$$ −1.92997 −0.964986
$$5$$ 1.43515 0.641820 0.320910 0.947110i $$-0.396011\pi$$
0.320910 + 0.947110i $$0.396011\pi$$
$$6$$ −0.769807 −0.314273
$$7$$ 0 0
$$8$$ 1.03998 0.367687
$$9$$ 5.46247 1.82082
$$10$$ −0.379780 −0.120097
$$11$$ −5.50474 −1.65974 −0.829871 0.557955i $$-0.811586\pi$$
−0.829871 + 0.557955i $$0.811586\pi$$
$$12$$ −5.61435 −1.62072
$$13$$ 0 0
$$14$$ 0 0
$$15$$ 4.17491 1.07796
$$16$$ 3.58474 0.896185
$$17$$ 4.83072 1.17162 0.585811 0.810448i $$-0.300776\pi$$
0.585811 + 0.810448i $$0.300776\pi$$
$$18$$ −1.44552 −0.340711
$$19$$ −2.82036 −0.647035 −0.323518 0.946222i $$-0.604865\pi$$
−0.323518 + 0.946222i $$0.604865\pi$$
$$20$$ −2.76981 −0.619348
$$21$$ 0 0
$$22$$ 1.45670 0.310570
$$23$$ −5.99956 −1.25100 −0.625498 0.780226i $$-0.715104\pi$$
−0.625498 + 0.780226i $$0.715104\pi$$
$$24$$ 3.02532 0.617541
$$25$$ −2.94033 −0.588067
$$26$$ 0 0
$$27$$ 7.16341 1.37860
$$28$$ 0 0
$$29$$ 1.04188 0.193472 0.0967361 0.995310i $$-0.469160\pi$$
0.0967361 + 0.995310i $$0.469160\pi$$
$$30$$ −1.10479 −0.201706
$$31$$ −9.20895 −1.65398 −0.826988 0.562219i $$-0.809948\pi$$
−0.826988 + 0.562219i $$0.809948\pi$$
$$32$$ −3.02857 −0.535380
$$33$$ −16.0135 −2.78759
$$34$$ −1.27834 −0.219233
$$35$$ 0 0
$$36$$ −10.5424 −1.75707
$$37$$ −0.612497 −0.100694 −0.0503470 0.998732i $$-0.516033\pi$$
−0.0503470 + 0.998732i $$0.516033\pi$$
$$38$$ 0.746342 0.121073
$$39$$ 0 0
$$40$$ 1.49252 0.235989
$$41$$ 10.6196 1.65850 0.829249 0.558879i $$-0.188768\pi$$
0.829249 + 0.558879i $$0.188768\pi$$
$$42$$ 0 0
$$43$$ −8.43685 −1.28661 −0.643304 0.765611i $$-0.722437\pi$$
−0.643304 + 0.765611i $$0.722437\pi$$
$$44$$ 10.6240 1.60163
$$45$$ 7.83949 1.16864
$$46$$ 1.58764 0.234085
$$47$$ −2.40922 −0.351422 −0.175711 0.984442i $$-0.556222\pi$$
−0.175711 + 0.984442i $$0.556222\pi$$
$$48$$ 10.4281 1.50517
$$49$$ 0 0
$$50$$ 0.778091 0.110039
$$51$$ 14.0527 1.96778
$$52$$ 0 0
$$53$$ −1.82959 −0.251313 −0.125657 0.992074i $$-0.540104\pi$$
−0.125657 + 0.992074i $$0.540104\pi$$
$$54$$ −1.89563 −0.257962
$$55$$ −7.90015 −1.06526
$$56$$ 0 0
$$57$$ −8.20452 −1.08672
$$58$$ −0.275709 −0.0362024
$$59$$ 0.870914 0.113383 0.0566917 0.998392i $$-0.481945\pi$$
0.0566917 + 0.998392i $$0.481945\pi$$
$$60$$ −8.05746 −1.04021
$$61$$ −3.33253 −0.426686 −0.213343 0.976977i $$-0.568435\pi$$
−0.213343 + 0.976977i $$0.568435\pi$$
$$62$$ 2.43693 0.309491
$$63$$ 0 0
$$64$$ −6.36804 −0.796005
$$65$$ 0 0
$$66$$ 4.23759 0.521611
$$67$$ 6.62741 0.809667 0.404833 0.914390i $$-0.367329\pi$$
0.404833 + 0.914390i $$0.367329\pi$$
$$68$$ −9.32316 −1.13060
$$69$$ −17.4529 −2.10108
$$70$$ 0 0
$$71$$ 6.85856 0.813961 0.406980 0.913437i $$-0.366582\pi$$
0.406980 + 0.913437i $$0.366582\pi$$
$$72$$ 5.68083 0.669493
$$73$$ −3.14147 −0.367682 −0.183841 0.982956i $$-0.558853\pi$$
−0.183841 + 0.982956i $$0.558853\pi$$
$$74$$ 0.162083 0.0188418
$$75$$ −8.55353 −0.987676
$$76$$ 5.44322 0.624380
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −17.5723 −1.97704 −0.988518 0.151101i $$-0.951718\pi$$
−0.988518 + 0.151101i $$0.951718\pi$$
$$80$$ 5.14465 0.575190
$$81$$ 4.45118 0.494576
$$82$$ −2.81022 −0.310337
$$83$$ −11.4525 −1.25708 −0.628538 0.777779i $$-0.716346\pi$$
−0.628538 + 0.777779i $$0.716346\pi$$
$$84$$ 0 0
$$85$$ 6.93283 0.751971
$$86$$ 2.23261 0.240749
$$87$$ 3.03086 0.324943
$$88$$ −5.72479 −0.610265
$$89$$ 0.995318 0.105503 0.0527517 0.998608i $$-0.483201\pi$$
0.0527517 + 0.998608i $$0.483201\pi$$
$$90$$ −2.07454 −0.218675
$$91$$ 0 0
$$92$$ 11.5790 1.20719
$$93$$ −26.7891 −2.77790
$$94$$ 0.637545 0.0657577
$$95$$ −4.04765 −0.415280
$$96$$ −8.81020 −0.899188
$$97$$ −13.5090 −1.37163 −0.685817 0.727774i $$-0.740555\pi$$
−0.685817 + 0.727774i $$0.740555\pi$$
$$98$$ 0 0
$$99$$ −30.0695 −3.02210
$$100$$ 5.67477 0.567477
$$101$$ −1.00807 −0.100306 −0.0501532 0.998742i $$-0.515971\pi$$
−0.0501532 + 0.998742i $$0.515971\pi$$
$$102$$ −3.71873 −0.368209
$$103$$ −12.7754 −1.25880 −0.629401 0.777081i $$-0.716700\pi$$
−0.629401 + 0.777081i $$0.716700\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0.484157 0.0470255
$$107$$ −0.685495 −0.0662693 −0.0331347 0.999451i $$-0.510549\pi$$
−0.0331347 + 0.999451i $$0.510549\pi$$
$$108$$ −13.8252 −1.33033
$$109$$ 2.90344 0.278099 0.139050 0.990285i $$-0.455595\pi$$
0.139050 + 0.990285i $$0.455595\pi$$
$$110$$ 2.09059 0.199330
$$111$$ −1.78177 −0.169119
$$112$$ 0 0
$$113$$ 12.0315 1.13183 0.565915 0.824464i $$-0.308523\pi$$
0.565915 + 0.824464i $$0.308523\pi$$
$$114$$ 2.17113 0.203345
$$115$$ −8.61029 −0.802914
$$116$$ −2.01080 −0.186698
$$117$$ 0 0
$$118$$ −0.230467 −0.0212162
$$119$$ 0 0
$$120$$ 4.34180 0.396350
$$121$$ 19.3022 1.75474
$$122$$ 0.881875 0.0798412
$$123$$ 30.8927 2.78550
$$124$$ 17.7730 1.59606
$$125$$ −11.3956 −1.01925
$$126$$ 0 0
$$127$$ 15.6659 1.39012 0.695062 0.718950i $$-0.255377\pi$$
0.695062 + 0.718950i $$0.255377\pi$$
$$128$$ 7.74229 0.684328
$$129$$ −24.5431 −2.16090
$$130$$ 0 0
$$131$$ 12.1273 1.05957 0.529784 0.848132i $$-0.322273\pi$$
0.529784 + 0.848132i $$0.322273\pi$$
$$132$$ 30.9056 2.68998
$$133$$ 0 0
$$134$$ −1.75379 −0.151504
$$135$$ 10.2806 0.884813
$$136$$ 5.02383 0.430790
$$137$$ −15.9375 −1.36163 −0.680815 0.732456i $$-0.738374\pi$$
−0.680815 + 0.732456i $$0.738374\pi$$
$$138$$ 4.61851 0.393153
$$139$$ −6.64088 −0.563272 −0.281636 0.959521i $$-0.590877\pi$$
−0.281636 + 0.959521i $$0.590877\pi$$
$$140$$ 0 0
$$141$$ −7.00851 −0.590223
$$142$$ −1.81496 −0.152308
$$143$$ 0 0
$$144$$ 19.5815 1.63180
$$145$$ 1.49526 0.124174
$$146$$ 0.831317 0.0688003
$$147$$ 0 0
$$148$$ 1.18210 0.0971683
$$149$$ 19.5502 1.60162 0.800809 0.598920i $$-0.204403\pi$$
0.800809 + 0.598920i $$0.204403\pi$$
$$150$$ 2.26349 0.184813
$$151$$ −10.6880 −0.869779 −0.434890 0.900484i $$-0.643213\pi$$
−0.434890 + 0.900484i $$0.643213\pi$$
$$152$$ −2.93311 −0.237906
$$153$$ 26.3877 2.13332
$$154$$ 0 0
$$155$$ −13.2163 −1.06156
$$156$$ 0 0
$$157$$ −15.0734 −1.20299 −0.601496 0.798876i $$-0.705428\pi$$
−0.601496 + 0.798876i $$0.705428\pi$$
$$158$$ 4.65009 0.369942
$$159$$ −5.32233 −0.422088
$$160$$ −4.34646 −0.343618
$$161$$ 0 0
$$162$$ −1.17790 −0.0925447
$$163$$ 23.8135 1.86521 0.932607 0.360894i $$-0.117528\pi$$
0.932607 + 0.360894i $$0.117528\pi$$
$$164$$ −20.4955 −1.60043
$$165$$ −22.9818 −1.78913
$$166$$ 3.03064 0.235223
$$167$$ 7.12371 0.551249 0.275625 0.961265i $$-0.411115\pi$$
0.275625 + 0.961265i $$0.411115\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ −1.83461 −0.140708
$$171$$ −15.4061 −1.17814
$$172$$ 16.2829 1.24156
$$173$$ 11.2367 0.854309 0.427155 0.904179i $$-0.359516\pi$$
0.427155 + 0.904179i $$0.359516\pi$$
$$174$$ −0.802047 −0.0608030
$$175$$ 0 0
$$176$$ −19.7331 −1.48744
$$177$$ 2.53352 0.190431
$$178$$ −0.263387 −0.0197417
$$179$$ −13.1945 −0.986204 −0.493102 0.869972i $$-0.664137\pi$$
−0.493102 + 0.869972i $$0.664137\pi$$
$$180$$ −15.1300 −1.12772
$$181$$ −13.7414 −1.02139 −0.510696 0.859761i $$-0.670612\pi$$
−0.510696 + 0.859761i $$0.670612\pi$$
$$182$$ 0 0
$$183$$ −9.69443 −0.716633
$$184$$ −6.23939 −0.459974
$$185$$ −0.879028 −0.0646274
$$186$$ 7.08912 0.519799
$$187$$ −26.5919 −1.94459
$$188$$ 4.64974 0.339117
$$189$$ 0 0
$$190$$ 1.07112 0.0777069
$$191$$ −16.3307 −1.18165 −0.590824 0.806800i $$-0.701197\pi$$
−0.590824 + 0.806800i $$0.701197\pi$$
$$192$$ −18.5248 −1.33692
$$193$$ 14.0533 1.01158 0.505790 0.862656i $$-0.331201\pi$$
0.505790 + 0.862656i $$0.331201\pi$$
$$194$$ 3.57485 0.256659
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.46898 0.104660 0.0523302 0.998630i $$-0.483335\pi$$
0.0523302 + 0.998630i $$0.483335\pi$$
$$198$$ 7.95719 0.565493
$$199$$ −13.3772 −0.948285 −0.474142 0.880448i $$-0.657242\pi$$
−0.474142 + 0.880448i $$0.657242\pi$$
$$200$$ −3.05787 −0.216224
$$201$$ 19.2794 1.35986
$$202$$ 0.266761 0.0187692
$$203$$ 0 0
$$204$$ −27.1214 −1.89888
$$205$$ 15.2407 1.06446
$$206$$ 3.38072 0.235546
$$207$$ −32.7724 −2.27784
$$208$$ 0 0
$$209$$ 15.5254 1.07391
$$210$$ 0 0
$$211$$ 3.47044 0.238915 0.119457 0.992839i $$-0.461885\pi$$
0.119457 + 0.992839i $$0.461885\pi$$
$$212$$ 3.53105 0.242514
$$213$$ 19.9518 1.36707
$$214$$ 0.181400 0.0124003
$$215$$ −12.1082 −0.825771
$$216$$ 7.44977 0.506893
$$217$$ 0 0
$$218$$ −0.768328 −0.0520378
$$219$$ −9.13865 −0.617533
$$220$$ 15.2471 1.02796
$$221$$ 0 0
$$222$$ 0.471505 0.0316453
$$223$$ 9.91318 0.663836 0.331918 0.943308i $$-0.392304\pi$$
0.331918 + 0.943308i $$0.392304\pi$$
$$224$$ 0 0
$$225$$ −16.0615 −1.07077
$$226$$ −3.18386 −0.211787
$$227$$ −12.0727 −0.801292 −0.400646 0.916233i $$-0.631214\pi$$
−0.400646 + 0.916233i $$0.631214\pi$$
$$228$$ 15.8345 1.04867
$$229$$ −4.05171 −0.267745 −0.133872 0.990999i $$-0.542741\pi$$
−0.133872 + 0.990999i $$0.542741\pi$$
$$230$$ 2.27851 0.150241
$$231$$ 0 0
$$232$$ 1.08353 0.0711372
$$233$$ 12.5450 0.821850 0.410925 0.911669i $$-0.365206\pi$$
0.410925 + 0.911669i $$0.365206\pi$$
$$234$$ 0 0
$$235$$ −3.45761 −0.225549
$$236$$ −1.68084 −0.109413
$$237$$ −51.1184 −3.32049
$$238$$ 0 0
$$239$$ −13.3463 −0.863299 −0.431649 0.902042i $$-0.642068\pi$$
−0.431649 + 0.902042i $$0.642068\pi$$
$$240$$ 14.9660 0.966049
$$241$$ −20.3854 −1.31314 −0.656568 0.754267i $$-0.727993\pi$$
−0.656568 + 0.754267i $$0.727993\pi$$
$$242$$ −5.10787 −0.328346
$$243$$ −8.54160 −0.547944
$$244$$ 6.43169 0.411747
$$245$$ 0 0
$$246$$ −8.17502 −0.521221
$$247$$ 0 0
$$248$$ −9.57708 −0.608145
$$249$$ −33.3157 −2.11130
$$250$$ 3.01558 0.190722
$$251$$ 17.1921 1.08515 0.542577 0.840006i $$-0.317449\pi$$
0.542577 + 0.840006i $$0.317449\pi$$
$$252$$ 0 0
$$253$$ 33.0260 2.07633
$$254$$ −4.14561 −0.260119
$$255$$ 20.1678 1.26296
$$256$$ 10.6873 0.667954
$$257$$ −7.64695 −0.477004 −0.238502 0.971142i $$-0.576656\pi$$
−0.238502 + 0.971142i $$0.576656\pi$$
$$258$$ 6.49475 0.404345
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 5.69124 0.352279
$$262$$ −3.20921 −0.198266
$$263$$ −0.101037 −0.00623022 −0.00311511 0.999995i $$-0.500992\pi$$
−0.00311511 + 0.999995i $$0.500992\pi$$
$$264$$ −16.6536 −1.02496
$$265$$ −2.62574 −0.161298
$$266$$ 0 0
$$267$$ 2.89541 0.177196
$$268$$ −12.7907 −0.781318
$$269$$ −7.56852 −0.461461 −0.230730 0.973018i $$-0.574111\pi$$
−0.230730 + 0.973018i $$0.574111\pi$$
$$270$$ −2.72052 −0.165565
$$271$$ 13.8554 0.841653 0.420826 0.907141i $$-0.361740\pi$$
0.420826 + 0.907141i $$0.361740\pi$$
$$272$$ 17.3169 1.04999
$$273$$ 0 0
$$274$$ 4.21748 0.254787
$$275$$ 16.1858 0.976039
$$276$$ 33.6837 2.02752
$$277$$ 0.552935 0.0332226 0.0166113 0.999862i $$-0.494712\pi$$
0.0166113 + 0.999862i $$0.494712\pi$$
$$278$$ 1.75735 0.105399
$$279$$ −50.3036 −3.01160
$$280$$ 0 0
$$281$$ −1.14667 −0.0684043 −0.0342022 0.999415i $$-0.510889\pi$$
−0.0342022 + 0.999415i $$0.510889\pi$$
$$282$$ 1.85464 0.110442
$$283$$ −4.05396 −0.240983 −0.120491 0.992714i $$-0.538447\pi$$
−0.120491 + 0.992714i $$0.538447\pi$$
$$284$$ −13.2368 −0.785461
$$285$$ −11.7747 −0.697476
$$286$$ 0 0
$$287$$ 0 0
$$288$$ −16.5435 −0.974833
$$289$$ 6.33588 0.372699
$$290$$ −0.395685 −0.0232354
$$291$$ −39.2982 −2.30370
$$292$$ 6.06296 0.354808
$$293$$ −15.0649 −0.880102 −0.440051 0.897973i $$-0.645040\pi$$
−0.440051 + 0.897973i $$0.645040\pi$$
$$294$$ 0 0
$$295$$ 1.24990 0.0727718
$$296$$ −0.636982 −0.0370238
$$297$$ −39.4327 −2.28812
$$298$$ −5.17351 −0.299694
$$299$$ 0 0
$$300$$ 16.5081 0.953094
$$301$$ 0 0
$$302$$ 2.82834 0.162752
$$303$$ −2.93250 −0.168468
$$304$$ −10.1103 −0.579863
$$305$$ −4.78269 −0.273856
$$306$$ −6.98288 −0.399185
$$307$$ 19.9408 1.13808 0.569040 0.822310i $$-0.307315\pi$$
0.569040 + 0.822310i $$0.307315\pi$$
$$308$$ 0 0
$$309$$ −37.1642 −2.11420
$$310$$ 3.49737 0.198637
$$311$$ −10.8956 −0.617833 −0.308916 0.951089i $$-0.599966\pi$$
−0.308916 + 0.951089i $$0.599966\pi$$
$$312$$ 0 0
$$313$$ −0.0519190 −0.00293464 −0.00146732 0.999999i $$-0.500467\pi$$
−0.00146732 + 0.999999i $$0.500467\pi$$
$$314$$ 3.98883 0.225103
$$315$$ 0 0
$$316$$ 33.9140 1.90781
$$317$$ −16.1010 −0.904321 −0.452161 0.891937i $$-0.649347\pi$$
−0.452161 + 0.891937i $$0.649347\pi$$
$$318$$ 1.40843 0.0789808
$$319$$ −5.73528 −0.321114
$$320$$ −9.13912 −0.510892
$$321$$ −1.99413 −0.111301
$$322$$ 0 0
$$323$$ −13.6244 −0.758081
$$324$$ −8.59066 −0.477259
$$325$$ 0 0
$$326$$ −6.30167 −0.349017
$$327$$ 8.44621 0.467077
$$328$$ 11.0441 0.609808
$$329$$ 0 0
$$330$$ 6.08159 0.334781
$$331$$ −30.7862 −1.69216 −0.846081 0.533054i $$-0.821044\pi$$
−0.846081 + 0.533054i $$0.821044\pi$$
$$332$$ 22.1030 1.21306
$$333$$ −3.34575 −0.183346
$$334$$ −1.88512 −0.103149
$$335$$ 9.51135 0.519661
$$336$$ 0 0
$$337$$ −2.41842 −0.131740 −0.0658700 0.997828i $$-0.520982\pi$$
−0.0658700 + 0.997828i $$0.520982\pi$$
$$338$$ 0 0
$$339$$ 35.0001 1.90094
$$340$$ −13.3802 −0.725642
$$341$$ 50.6929 2.74517
$$342$$ 4.07687 0.220452
$$343$$ 0 0
$$344$$ −8.77411 −0.473069
$$345$$ −25.0476 −1.34852
$$346$$ −2.97353 −0.159858
$$347$$ −0.492527 −0.0264403 −0.0132201 0.999913i $$-0.504208\pi$$
−0.0132201 + 0.999913i $$0.504208\pi$$
$$348$$ −5.84948 −0.313565
$$349$$ 11.9442 0.639356 0.319678 0.947526i $$-0.396425\pi$$
0.319678 + 0.947526i $$0.396425\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 16.6715 0.888593
$$353$$ −15.5299 −0.826575 −0.413288 0.910601i $$-0.635620\pi$$
−0.413288 + 0.910601i $$0.635620\pi$$
$$354$$ −0.670436 −0.0356333
$$355$$ 9.84308 0.522417
$$356$$ −1.92094 −0.101809
$$357$$ 0 0
$$358$$ 3.49162 0.184538
$$359$$ 8.50709 0.448987 0.224493 0.974476i $$-0.427927\pi$$
0.224493 + 0.974476i $$0.427927\pi$$
$$360$$ 8.15287 0.429694
$$361$$ −11.0456 −0.581345
$$362$$ 3.63635 0.191122
$$363$$ 56.1507 2.94715
$$364$$ 0 0
$$365$$ −4.50850 −0.235985
$$366$$ 2.56540 0.134096
$$367$$ 5.19084 0.270960 0.135480 0.990780i $$-0.456742\pi$$
0.135480 + 0.990780i $$0.456742\pi$$
$$368$$ −21.5069 −1.12112
$$369$$ 58.0091 3.01983
$$370$$ 0.232614 0.0120930
$$371$$ 0 0
$$372$$ 51.7023 2.68064
$$373$$ −10.1427 −0.525169 −0.262585 0.964909i $$-0.584575\pi$$
−0.262585 + 0.964909i $$0.584575\pi$$
$$374$$ 7.03692 0.363870
$$375$$ −33.1502 −1.71187
$$376$$ −2.50553 −0.129213
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 3.63670 0.186805 0.0934024 0.995628i $$-0.470226\pi$$
0.0934024 + 0.995628i $$0.470226\pi$$
$$380$$ 7.81186 0.400740
$$381$$ 45.5726 2.33476
$$382$$ 4.32154 0.221109
$$383$$ −4.60281 −0.235192 −0.117596 0.993061i $$-0.537519\pi$$
−0.117596 + 0.993061i $$0.537519\pi$$
$$384$$ 22.5226 1.14935
$$385$$ 0 0
$$386$$ −3.71888 −0.189286
$$387$$ −46.0861 −2.34269
$$388$$ 26.0721 1.32361
$$389$$ 19.6104 0.994286 0.497143 0.867669i $$-0.334382\pi$$
0.497143 + 0.867669i $$0.334382\pi$$
$$390$$ 0 0
$$391$$ −28.9822 −1.46569
$$392$$ 0 0
$$393$$ 35.2788 1.77958
$$394$$ −0.388731 −0.0195840
$$395$$ −25.2189 −1.26890
$$396$$ 58.0333 2.91628
$$397$$ 19.8635 0.996919 0.498459 0.866913i $$-0.333899\pi$$
0.498459 + 0.866913i $$0.333899\pi$$
$$398$$ 3.53996 0.177442
$$399$$ 0 0
$$400$$ −10.5403 −0.527017
$$401$$ 15.1117 0.754644 0.377322 0.926082i $$-0.376845\pi$$
0.377322 + 0.926082i $$0.376845\pi$$
$$402$$ −5.10183 −0.254456
$$403$$ 0 0
$$404$$ 1.94554 0.0967943
$$405$$ 6.38813 0.317429
$$406$$ 0 0
$$407$$ 3.37164 0.167126
$$408$$ 14.6145 0.723525
$$409$$ 35.2443 1.74272 0.871360 0.490644i $$-0.163238\pi$$
0.871360 + 0.490644i $$0.163238\pi$$
$$410$$ −4.03310 −0.199181
$$411$$ −46.3626 −2.28690
$$412$$ 24.6563 1.21473
$$413$$ 0 0
$$414$$ 8.67246 0.426228
$$415$$ −16.4361 −0.806816
$$416$$ 0 0
$$417$$ −19.3185 −0.946033
$$418$$ −4.10842 −0.200950
$$419$$ 1.50468 0.0735084 0.0367542 0.999324i $$-0.488298\pi$$
0.0367542 + 0.999324i $$0.488298\pi$$
$$420$$ 0 0
$$421$$ 24.5079 1.19444 0.597221 0.802077i $$-0.296272\pi$$
0.597221 + 0.802077i $$0.296272\pi$$
$$422$$ −0.918370 −0.0447056
$$423$$ −13.1603 −0.639877
$$424$$ −1.90272 −0.0924045
$$425$$ −14.2039 −0.688992
$$426$$ −5.27977 −0.255806
$$427$$ 0 0
$$428$$ 1.32299 0.0639490
$$429$$ 0 0
$$430$$ 3.20414 0.154518
$$431$$ −41.0655 −1.97805 −0.989027 0.147732i $$-0.952803\pi$$
−0.989027 + 0.147732i $$0.952803\pi$$
$$432$$ 25.6790 1.23548
$$433$$ 6.65603 0.319869 0.159934 0.987128i $$-0.448872\pi$$
0.159934 + 0.987128i $$0.448872\pi$$
$$434$$ 0 0
$$435$$ 4.34975 0.208555
$$436$$ −5.60357 −0.268362
$$437$$ 16.9209 0.809438
$$438$$ 2.41833 0.115552
$$439$$ 8.22990 0.392792 0.196396 0.980525i $$-0.437076\pi$$
0.196396 + 0.980525i $$0.437076\pi$$
$$440$$ −8.21596 −0.391681
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 17.6856 0.840266 0.420133 0.907463i $$-0.361983\pi$$
0.420133 + 0.907463i $$0.361983\pi$$
$$444$$ 3.43878 0.163197
$$445$$ 1.42843 0.0677143
$$446$$ −2.62329 −0.124216
$$447$$ 56.8723 2.68997
$$448$$ 0 0
$$449$$ 14.5250 0.685477 0.342738 0.939431i $$-0.388646\pi$$
0.342738 + 0.939431i $$0.388646\pi$$
$$450$$ 4.25030 0.200361
$$451$$ −58.4580 −2.75268
$$452$$ −23.2205 −1.09220
$$453$$ −31.0918 −1.46082
$$454$$ 3.19475 0.149937
$$455$$ 0 0
$$456$$ −8.53250 −0.399571
$$457$$ −3.78919 −0.177251 −0.0886255 0.996065i $$-0.528247\pi$$
−0.0886255 + 0.996065i $$0.528247\pi$$
$$458$$ 1.07219 0.0501002
$$459$$ 34.6045 1.61520
$$460$$ 16.6176 0.774801
$$461$$ −13.1107 −0.610627 −0.305314 0.952252i $$-0.598761\pi$$
−0.305314 + 0.952252i $$0.598761\pi$$
$$462$$ 0 0
$$463$$ −15.3027 −0.711176 −0.355588 0.934643i $$-0.615719\pi$$
−0.355588 + 0.934643i $$0.615719\pi$$
$$464$$ 3.73487 0.173387
$$465$$ −38.4465 −1.78292
$$466$$ −3.31974 −0.153784
$$467$$ 30.7738 1.42404 0.712022 0.702158i $$-0.247780\pi$$
0.712022 + 0.702158i $$0.247780\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0.914975 0.0422046
$$471$$ −43.8491 −2.02046
$$472$$ 0.905729 0.0416896
$$473$$ 46.4427 2.13544
$$474$$ 13.5273 0.621328
$$475$$ 8.29280 0.380500
$$476$$ 0 0
$$477$$ −9.99407 −0.457597
$$478$$ 3.53178 0.161540
$$479$$ 1.71740 0.0784699 0.0392350 0.999230i $$-0.487508\pi$$
0.0392350 + 0.999230i $$0.487508\pi$$
$$480$$ −12.6440 −0.577117
$$481$$ 0 0
$$482$$ 5.39451 0.245713
$$483$$ 0 0
$$484$$ −37.2527 −1.69330
$$485$$ −19.3875 −0.880342
$$486$$ 2.26033 0.102531
$$487$$ −22.6805 −1.02775 −0.513877 0.857864i $$-0.671791\pi$$
−0.513877 + 0.857864i $$0.671791\pi$$
$$488$$ −3.46575 −0.156887
$$489$$ 69.2741 3.13268
$$490$$ 0 0
$$491$$ −13.1366 −0.592846 −0.296423 0.955057i $$-0.595794\pi$$
−0.296423 + 0.955057i $$0.595794\pi$$
$$492$$ −59.6220 −2.68797
$$493$$ 5.03303 0.226676
$$494$$ 0 0
$$495$$ −43.1543 −1.93964
$$496$$ −33.0117 −1.48227
$$497$$ 0 0
$$498$$ 8.81622 0.395064
$$499$$ −14.0395 −0.628495 −0.314248 0.949341i $$-0.601752\pi$$
−0.314248 + 0.949341i $$0.601752\pi$$
$$500$$ 21.9932 0.983566
$$501$$ 20.7231 0.925840
$$502$$ −4.54948 −0.203053
$$503$$ 0.367865 0.0164023 0.00820114 0.999966i $$-0.497389\pi$$
0.00820114 + 0.999966i $$0.497389\pi$$
$$504$$ 0 0
$$505$$ −1.44673 −0.0643786
$$506$$ −8.73957 −0.388521
$$507$$ 0 0
$$508$$ −30.2348 −1.34145
$$509$$ −41.2319 −1.82757 −0.913787 0.406194i $$-0.866856\pi$$
−0.913787 + 0.406194i $$0.866856\pi$$
$$510$$ −5.33694 −0.236324
$$511$$ 0 0
$$512$$ −18.3127 −0.809315
$$513$$ −20.2034 −0.892002
$$514$$ 2.02359 0.0892566
$$515$$ −18.3347 −0.807925
$$516$$ 47.3675 2.08524
$$517$$ 13.2622 0.583269
$$518$$ 0 0
$$519$$ 32.6879 1.43484
$$520$$ 0 0
$$521$$ 1.04099 0.0456065 0.0228032 0.999740i $$-0.492741\pi$$
0.0228032 + 0.999740i $$0.492741\pi$$
$$522$$ −1.50605 −0.0659182
$$523$$ −20.0209 −0.875451 −0.437726 0.899109i $$-0.644216\pi$$
−0.437726 + 0.899109i $$0.644216\pi$$
$$524$$ −23.4054 −1.02247
$$525$$ 0 0
$$526$$ 0.0267371 0.00116579
$$527$$ −44.4859 −1.93784
$$528$$ −57.4042 −2.49820
$$529$$ 12.9947 0.564989
$$530$$ 0.694840 0.0301819
$$531$$ 4.75735 0.206451
$$532$$ 0 0
$$533$$ 0 0
$$534$$ −0.766203 −0.0331568
$$535$$ −0.983791 −0.0425330
$$536$$ 6.89234 0.297704
$$537$$ −38.3833 −1.65636
$$538$$ 2.00283 0.0863481
$$539$$ 0 0
$$540$$ −19.8413 −0.853832
$$541$$ 9.78749 0.420797 0.210399 0.977616i $$-0.432524\pi$$
0.210399 + 0.977616i $$0.432524\pi$$
$$542$$ −3.66649 −0.157489
$$543$$ −39.9743 −1.71546
$$544$$ −14.6302 −0.627263
$$545$$ 4.16689 0.178490
$$546$$ 0 0
$$547$$ −2.56174 −0.109532 −0.0547660 0.998499i $$-0.517441\pi$$
−0.0547660 + 0.998499i $$0.517441\pi$$
$$548$$ 30.7589 1.31395
$$549$$ −18.2038 −0.776921
$$550$$ −4.28319 −0.182636
$$551$$ −2.93848 −0.125183
$$552$$ −18.1506 −0.772541
$$553$$ 0 0
$$554$$ −0.146321 −0.00621660
$$555$$ −2.55712 −0.108544
$$556$$ 12.8167 0.543550
$$557$$ 27.4442 1.16285 0.581424 0.813601i $$-0.302496\pi$$
0.581424 + 0.813601i $$0.302496\pi$$
$$558$$ 13.3117 0.563528
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −77.3567 −3.26600
$$562$$ 0.303438 0.0127998
$$563$$ 0.162708 0.00685734 0.00342867 0.999994i $$-0.498909\pi$$
0.00342867 + 0.999994i $$0.498909\pi$$
$$564$$ 13.5262 0.569557
$$565$$ 17.2671 0.726431
$$566$$ 1.07278 0.0450925
$$567$$ 0 0
$$568$$ 7.13273 0.299283
$$569$$ −12.3901 −0.519419 −0.259709 0.965687i $$-0.583627\pi$$
−0.259709 + 0.965687i $$0.583627\pi$$
$$570$$ 3.11591 0.130511
$$571$$ −21.8122 −0.912810 −0.456405 0.889772i $$-0.650863\pi$$
−0.456405 + 0.889772i $$0.650863\pi$$
$$572$$ 0 0
$$573$$ −47.5066 −1.98462
$$574$$ 0 0
$$575$$ 17.6407 0.735669
$$576$$ −34.7852 −1.44939
$$577$$ −6.06583 −0.252524 −0.126262 0.991997i $$-0.540298\pi$$
−0.126262 + 0.991997i $$0.540298\pi$$
$$578$$ −1.67664 −0.0697391
$$579$$ 40.8816 1.69898
$$580$$ −2.88581 −0.119827
$$581$$ 0 0
$$582$$ 10.3993 0.431067
$$583$$ 10.0714 0.417115
$$584$$ −3.26705 −0.135192
$$585$$ 0 0
$$586$$ 3.98658 0.164684
$$587$$ −20.5820 −0.849510 −0.424755 0.905308i $$-0.639640\pi$$
−0.424755 + 0.905308i $$0.639640\pi$$
$$588$$ 0 0
$$589$$ 25.9726 1.07018
$$590$$ −0.330756 −0.0136170
$$591$$ 4.27331 0.175781
$$592$$ −2.19564 −0.0902404
$$593$$ 24.0397 0.987190 0.493595 0.869692i $$-0.335682\pi$$
0.493595 + 0.869692i $$0.335682\pi$$
$$594$$ 10.4349 0.428151
$$595$$ 0 0
$$596$$ −37.7314 −1.54554
$$597$$ −38.9147 −1.59267
$$598$$ 0 0
$$599$$ 32.2523 1.31779 0.658896 0.752234i $$-0.271024\pi$$
0.658896 + 0.752234i $$0.271024\pi$$
$$600$$ −8.89546 −0.363156
$$601$$ 5.21454 0.212705 0.106353 0.994328i $$-0.466083\pi$$
0.106353 + 0.994328i $$0.466083\pi$$
$$602$$ 0 0
$$603$$ 36.2020 1.47426
$$604$$ 20.6276 0.839325
$$605$$ 27.7016 1.12623
$$606$$ 0.776017 0.0315235
$$607$$ 9.07048 0.368160 0.184080 0.982911i $$-0.441070\pi$$
0.184080 + 0.982911i $$0.441070\pi$$
$$608$$ 8.54165 0.346410
$$609$$ 0 0
$$610$$ 1.26563 0.0512437
$$611$$ 0 0
$$612$$ −50.9275 −2.05862
$$613$$ −20.0920 −0.811507 −0.405754 0.913983i $$-0.632991\pi$$
−0.405754 + 0.913983i $$0.632991\pi$$
$$614$$ −5.27686 −0.212957
$$615$$ 44.3357 1.78779
$$616$$ 0 0
$$617$$ 12.9556 0.521572 0.260786 0.965397i $$-0.416018\pi$$
0.260786 + 0.965397i $$0.416018\pi$$
$$618$$ 9.83463 0.395607
$$619$$ 44.3644 1.78316 0.891578 0.452866i $$-0.149598\pi$$
0.891578 + 0.452866i $$0.149598\pi$$
$$620$$ 25.5070 1.02439
$$621$$ −42.9773 −1.72462
$$622$$ 2.88327 0.115608
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −1.65276 −0.0661105
$$626$$ 0.0137392 0.000549127 0
$$627$$ 45.1638 1.80367
$$628$$ 29.0913 1.16087
$$629$$ −2.95880 −0.117975
$$630$$ 0 0
$$631$$ −6.61717 −0.263426 −0.131713 0.991288i $$-0.542048\pi$$
−0.131713 + 0.991288i $$0.542048\pi$$
$$632$$ −18.2747 −0.726930
$$633$$ 10.0956 0.401265
$$634$$ 4.26075 0.169216
$$635$$ 22.4830 0.892210
$$636$$ 10.2719 0.407309
$$637$$ 0 0
$$638$$ 1.51771 0.0600866
$$639$$ 37.4647 1.48208
$$640$$ 11.1114 0.439216
$$641$$ 18.9567 0.748744 0.374372 0.927279i $$-0.377858\pi$$
0.374372 + 0.927279i $$0.377858\pi$$
$$642$$ 0.527699 0.0208266
$$643$$ −13.4019 −0.528517 −0.264259 0.964452i $$-0.585127\pi$$
−0.264259 + 0.964452i $$0.585127\pi$$
$$644$$ 0 0
$$645$$ −35.2231 −1.38691
$$646$$ 3.60537 0.141851
$$647$$ 42.7588 1.68102 0.840511 0.541794i $$-0.182255\pi$$
0.840511 + 0.541794i $$0.182255\pi$$
$$648$$ 4.62912 0.181849
$$649$$ −4.79416 −0.188187
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −45.9593 −1.79991
$$653$$ 10.9852 0.429884 0.214942 0.976627i $$-0.431044\pi$$
0.214942 + 0.976627i $$0.431044\pi$$
$$654$$ −2.23509 −0.0873990
$$655$$ 17.4046 0.680052
$$656$$ 38.0684 1.48632
$$657$$ −17.1602 −0.669483
$$658$$ 0 0
$$659$$ −17.7614 −0.691884 −0.345942 0.938256i $$-0.612441\pi$$
−0.345942 + 0.938256i $$0.612441\pi$$
$$660$$ 44.3542 1.72649
$$661$$ −8.18255 −0.318264 −0.159132 0.987257i $$-0.550870\pi$$
−0.159132 + 0.987257i $$0.550870\pi$$
$$662$$ 8.14685 0.316636
$$663$$ 0 0
$$664$$ −11.9103 −0.462210
$$665$$ 0 0
$$666$$ 0.885374 0.0343076
$$667$$ −6.25082 −0.242033
$$668$$ −13.7486 −0.531948
$$669$$ 28.8378 1.11493
$$670$$ −2.51696 −0.0972385
$$671$$ 18.3447 0.708189
$$672$$ 0 0
$$673$$ 9.30129 0.358539 0.179269 0.983800i $$-0.442627\pi$$
0.179269 + 0.983800i $$0.442627\pi$$
$$674$$ 0.639979 0.0246511
$$675$$ −21.0628 −0.810708
$$676$$ 0 0
$$677$$ −41.1552 −1.58172 −0.790862 0.611994i $$-0.790367\pi$$
−0.790862 + 0.611994i $$0.790367\pi$$
$$678$$ −9.26195 −0.355703
$$679$$ 0 0
$$680$$ 7.20997 0.276490
$$681$$ −35.1198 −1.34579
$$682$$ −13.4147 −0.513675
$$683$$ −39.2842 −1.50317 −0.751583 0.659638i $$-0.770709\pi$$
−0.751583 + 0.659638i $$0.770709\pi$$
$$684$$ 29.7334 1.13689
$$685$$ −22.8727 −0.873921
$$686$$ 0 0
$$687$$ −11.7866 −0.449685
$$688$$ −30.2439 −1.15304
$$689$$ 0 0
$$690$$ 6.62827 0.252334
$$691$$ 3.03355 0.115402 0.0577009 0.998334i $$-0.481623\pi$$
0.0577009 + 0.998334i $$0.481623\pi$$
$$692$$ −21.6865 −0.824397
$$693$$ 0 0
$$694$$ 0.130336 0.00494748
$$695$$ −9.53069 −0.361520
$$696$$ 3.15202 0.119477
$$697$$ 51.3002 1.94313
$$698$$ −3.16074 −0.119636
$$699$$ 36.4938 1.38032
$$700$$ 0 0
$$701$$ −26.2320 −0.990767 −0.495384 0.868674i $$-0.664972\pi$$
−0.495384 + 0.868674i $$0.664972\pi$$
$$702$$ 0 0
$$703$$ 1.72746 0.0651525
$$704$$ 35.0544 1.32116
$$705$$ −10.0583 −0.378817
$$706$$ 4.10963 0.154668
$$707$$ 0 0
$$708$$ −4.88962 −0.183763
$$709$$ 7.87770 0.295853 0.147927 0.988998i $$-0.452740\pi$$
0.147927 + 0.988998i $$0.452740\pi$$
$$710$$ −2.60474 −0.0977542
$$711$$ −95.9881 −3.59984
$$712$$ 1.03511 0.0387922
$$713$$ 55.2497 2.06912
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 25.4650 0.951673
$$717$$ −38.8247 −1.44994
$$718$$ −2.25120 −0.0840141
$$719$$ 45.6656 1.70304 0.851519 0.524323i $$-0.175682\pi$$
0.851519 + 0.524323i $$0.175682\pi$$
$$720$$ 28.1025 1.04732
$$721$$ 0 0
$$722$$ 2.92295 0.108781
$$723$$ −59.3017 −2.20545
$$724$$ 26.5206 0.985630
$$725$$ −3.06348 −0.113775
$$726$$ −14.8590 −0.551468
$$727$$ 37.5947 1.39431 0.697155 0.716921i $$-0.254449\pi$$
0.697155 + 0.716921i $$0.254449\pi$$
$$728$$ 0 0
$$729$$ −38.2013 −1.41486
$$730$$ 1.19307 0.0441574
$$731$$ −40.7561 −1.50742
$$732$$ 18.7100 0.691541
$$733$$ −53.1810 −1.96429 −0.982143 0.188138i $$-0.939755\pi$$
−0.982143 + 0.188138i $$0.939755\pi$$
$$734$$ −1.37363 −0.0507018
$$735$$ 0 0
$$736$$ 18.1701 0.669758
$$737$$ −36.4822 −1.34384
$$738$$ −15.3508 −0.565069
$$739$$ −41.9633 −1.54364 −0.771822 0.635839i $$-0.780654\pi$$
−0.771822 + 0.635839i $$0.780654\pi$$
$$740$$ 1.69650 0.0623646
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 38.5424 1.41398 0.706991 0.707222i $$-0.250052\pi$$
0.706991 + 0.707222i $$0.250052\pi$$
$$744$$ −27.8600 −1.02140
$$745$$ 28.0576 1.02795
$$746$$ 2.68403 0.0982693
$$747$$ −62.5590 −2.28891
$$748$$ 51.3216 1.87650
$$749$$ 0 0
$$750$$ 8.77242 0.320323
$$751$$ 36.2434 1.32254 0.661270 0.750148i $$-0.270018\pi$$
0.661270 + 0.750148i $$0.270018\pi$$
$$752$$ −8.63644 −0.314939
$$753$$ 50.0123 1.82255
$$754$$ 0 0
$$755$$ −15.3390 −0.558242
$$756$$ 0 0
$$757$$ 19.4752 0.707837 0.353919 0.935276i $$-0.384849\pi$$
0.353919 + 0.935276i $$0.384849\pi$$
$$758$$ −0.962368 −0.0349548
$$759$$ 96.0738 3.48726
$$760$$ −4.20946 −0.152693
$$761$$ 51.9059 1.88159 0.940793 0.338981i $$-0.110082\pi$$
0.940793 + 0.338981i $$0.110082\pi$$
$$762$$ −12.0597 −0.436878
$$763$$ 0 0
$$764$$ 31.5178 1.14028
$$765$$ 37.8704 1.36921
$$766$$ 1.21803 0.0440090
$$767$$ 0 0
$$768$$ 31.0896 1.12185
$$769$$ 7.31376 0.263741 0.131870 0.991267i $$-0.457902\pi$$
0.131870 + 0.991267i $$0.457902\pi$$
$$770$$ 0 0
$$771$$ −22.2452 −0.801142
$$772$$ −27.1225 −0.976161
$$773$$ 14.1844 0.510178 0.255089 0.966918i $$-0.417895\pi$$
0.255089 + 0.966918i $$0.417895\pi$$
$$774$$ 12.1956 0.438362
$$775$$ 27.0774 0.972649
$$776$$ −14.0491 −0.504332
$$777$$ 0 0
$$778$$ −5.18943 −0.186050
$$779$$ −29.9510 −1.07311
$$780$$ 0 0
$$781$$ −37.7546 −1.35097
$$782$$ 7.66946 0.274259
$$783$$ 7.46341 0.266721
$$784$$ 0 0
$$785$$ −21.6327 −0.772104
$$786$$ −9.33569 −0.332993
$$787$$ −31.2777 −1.11493 −0.557465 0.830201i $$-0.688226\pi$$
−0.557465 + 0.830201i $$0.688226\pi$$
$$788$$ −2.83509 −0.100996
$$789$$ −0.293921 −0.0104639
$$790$$ 6.67360 0.237436
$$791$$ 0 0
$$792$$ −31.2715 −1.11119
$$793$$ 0 0
$$794$$ −5.25640 −0.186543
$$795$$ −7.63836 −0.270905
$$796$$ 25.8176 0.915082
$$797$$ 20.2422 0.717017 0.358509 0.933526i $$-0.383285\pi$$
0.358509 + 0.933526i $$0.383285\pi$$
$$798$$ 0 0
$$799$$ −11.6383 −0.411733
$$800$$ 8.90500 0.314839
$$801$$ 5.43689 0.192103
$$802$$ −3.99897 −0.141208
$$803$$ 17.2930 0.610257
$$804$$ −37.2086 −1.31225
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −22.0171 −0.775037
$$808$$ −1.04836 −0.0368813
$$809$$ 7.88265 0.277139 0.138570 0.990353i $$-0.455750\pi$$
0.138570 + 0.990353i $$0.455750\pi$$
$$810$$ −1.69047 −0.0593970
$$811$$ 5.99962 0.210675 0.105338 0.994437i $$-0.466408\pi$$
0.105338 + 0.994437i $$0.466408\pi$$
$$812$$ 0 0
$$813$$ 40.3057 1.41358
$$814$$ −0.892226 −0.0312725
$$815$$ 34.1760 1.19713
$$816$$ 50.3754 1.76349
$$817$$ 23.7950 0.832480
$$818$$ −9.32659 −0.326096
$$819$$ 0 0
$$820$$ −29.4142 −1.02719
$$821$$ 19.1692 0.669011 0.334505 0.942394i $$-0.391431\pi$$
0.334505 + 0.942394i $$0.391431\pi$$
$$822$$ 12.2688 0.427923
$$823$$ −30.3735 −1.05875 −0.529376 0.848387i $$-0.677574\pi$$
−0.529376 + 0.848387i $$0.677574\pi$$
$$824$$ −13.2861 −0.462845
$$825$$ 47.0850 1.63929
$$826$$ 0 0
$$827$$ 14.6870 0.510717 0.255359 0.966846i $$-0.417807\pi$$
0.255359 + 0.966846i $$0.417807\pi$$
$$828$$ 63.2499 2.19809
$$829$$ −34.9985 −1.21555 −0.607774 0.794110i $$-0.707938\pi$$
−0.607774 + 0.794110i $$0.707938\pi$$
$$830$$ 4.34943 0.150971
$$831$$ 1.60851 0.0557985
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 5.11220 0.177021
$$835$$ 10.2236 0.353803
$$836$$ −29.9635 −1.03631
$$837$$ −65.9675 −2.28017
$$838$$ −0.398178 −0.0137548
$$839$$ −27.6333 −0.954008 −0.477004 0.878901i $$-0.658277\pi$$
−0.477004 + 0.878901i $$0.658277\pi$$
$$840$$ 0 0
$$841$$ −27.9145 −0.962568
$$842$$ −6.48544 −0.223503
$$843$$ −3.33569 −0.114887
$$844$$ −6.69785 −0.230550
$$845$$ 0 0
$$846$$ 3.48257 0.119733
$$847$$ 0 0
$$848$$ −6.55859 −0.225223
$$849$$ −11.7931 −0.404738
$$850$$ 3.75874 0.128924
$$851$$ 3.67472 0.125968
$$852$$ −38.5064 −1.31921
$$853$$ −32.6336 −1.11735 −0.558676 0.829386i $$-0.688690\pi$$
−0.558676 + 0.829386i $$0.688690\pi$$
$$854$$ 0 0
$$855$$ −22.1102 −0.756152
$$856$$ −0.712898 −0.0243664
$$857$$ −18.8742 −0.644730 −0.322365 0.946616i $$-0.604478\pi$$
−0.322365 + 0.946616i $$0.604478\pi$$
$$858$$ 0 0
$$859$$ 15.8242 0.539915 0.269957 0.962872i $$-0.412990\pi$$
0.269957 + 0.962872i $$0.412990\pi$$
$$860$$ 23.3684 0.796857
$$861$$ 0 0
$$862$$ 10.8670 0.370132
$$863$$ −52.3212 −1.78104 −0.890518 0.454948i $$-0.849658\pi$$
−0.890518 + 0.454948i $$0.849658\pi$$
$$864$$ −21.6949 −0.738075
$$865$$ 16.1264 0.548313
$$866$$ −1.76136 −0.0598536
$$867$$ 18.4313 0.625959
$$868$$ 0 0
$$869$$ 96.7309 3.28137
$$870$$ −1.15106 −0.0390246
$$871$$ 0 0
$$872$$ 3.01951 0.102253
$$873$$ −73.7927 −2.49750
$$874$$ −4.47773 −0.151461
$$875$$ 0 0
$$876$$ 17.6373 0.595911
$$877$$ 54.0162 1.82400 0.911999 0.410193i $$-0.134539\pi$$
0.911999 + 0.410193i $$0.134539\pi$$
$$878$$ −2.17785 −0.0734989
$$879$$ −43.8243 −1.47816
$$880$$ −28.3200 −0.954667
$$881$$ −42.0823 −1.41779 −0.708895 0.705314i $$-0.750806\pi$$
−0.708895 + 0.705314i $$0.750806\pi$$
$$882$$ 0 0
$$883$$ 36.5314 1.22938 0.614689 0.788769i $$-0.289281\pi$$
0.614689 + 0.788769i $$0.289281\pi$$
$$884$$ 0 0
$$885$$ 3.63599 0.122222
$$886$$ −4.68007 −0.157230
$$887$$ −11.4648 −0.384950 −0.192475 0.981302i $$-0.561651\pi$$
−0.192475 + 0.981302i $$0.561651\pi$$
$$888$$ −1.85300 −0.0621827
$$889$$ 0 0
$$890$$ −0.378001 −0.0126706
$$891$$ −24.5026 −0.820869
$$892$$ −19.1322 −0.640592
$$893$$ 6.79488 0.227382
$$894$$ −15.0499 −0.503345
$$895$$ −18.9361 −0.632965
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −3.84370 −0.128266
$$899$$ −9.59462 −0.319999
$$900$$ 30.9982 1.03327
$$901$$ −8.83823 −0.294444
$$902$$ 15.4695 0.515079
$$903$$ 0 0
$$904$$ 12.5125 0.416159
$$905$$ −19.7211 −0.655550
$$906$$ 8.22772 0.273348
$$907$$ −5.04665 −0.167571 −0.0837856 0.996484i $$-0.526701\pi$$
−0.0837856 + 0.996484i $$0.526701\pi$$
$$908$$ 23.2999 0.773236
$$909$$ −5.50653 −0.182640
$$910$$ 0 0
$$911$$ 47.5236 1.57453 0.787263 0.616618i $$-0.211497\pi$$
0.787263 + 0.616618i $$0.211497\pi$$
$$912$$ −29.4111 −0.973898
$$913$$ 63.0431 2.08642
$$914$$ 1.00272 0.0331671
$$915$$ −13.9130 −0.459950
$$916$$ 7.81969 0.258370
$$917$$ 0 0
$$918$$ −9.15726 −0.302235
$$919$$ −45.6698 −1.50651 −0.753254 0.657730i $$-0.771517\pi$$
−0.753254 + 0.657730i $$0.771517\pi$$
$$920$$ −8.95449 −0.295221
$$921$$ 58.0084 1.91144
$$922$$ 3.46945 0.114260
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 1.80095 0.0592148
$$926$$ 4.04950 0.133075
$$927$$ −69.7855 −2.29206
$$928$$ −3.15540 −0.103581
$$929$$ 44.4449 1.45819 0.729095 0.684412i $$-0.239941\pi$$
0.729095 + 0.684412i $$0.239941\pi$$
$$930$$ 10.1740 0.333618
$$931$$ 0 0
$$932$$ −24.2115 −0.793074
$$933$$ −31.6957 −1.03767
$$934$$ −8.14357 −0.266466
$$935$$ −38.1634 −1.24808
$$936$$ 0 0
$$937$$ 46.9796 1.53476 0.767379 0.641194i $$-0.221561\pi$$
0.767379 + 0.641194i $$0.221561\pi$$
$$938$$ 0 0
$$939$$ −0.151034 −0.00492881
$$940$$ 6.67309 0.217652
$$941$$ −35.5654 −1.15940 −0.579699 0.814830i $$-0.696830\pi$$
−0.579699 + 0.814830i $$0.696830\pi$$
$$942$$ 11.6036 0.378067
$$943$$ −63.7128 −2.07477
$$944$$ 3.12200 0.101613
$$945$$ 0 0
$$946$$ −12.2900 −0.399581
$$947$$ 22.3592 0.726576 0.363288 0.931677i $$-0.381654\pi$$
0.363288 + 0.931677i $$0.381654\pi$$
$$948$$ 98.6570 3.20423
$$949$$ 0 0
$$950$$ −2.19450 −0.0711989
$$951$$ −46.8383 −1.51884
$$952$$ 0 0
$$953$$ −46.7684 −1.51498 −0.757488 0.652849i $$-0.773574\pi$$
−0.757488 + 0.652849i $$0.773574\pi$$
$$954$$ 2.64470 0.0856252
$$955$$ −23.4371 −0.758406
$$956$$ 25.7579 0.833071
$$957$$ −16.6841 −0.539321
$$958$$ −0.454469 −0.0146832
$$959$$ 0 0
$$960$$ −26.5860 −0.858059
$$961$$ 53.8048 1.73564
$$962$$ 0 0
$$963$$ −3.74450 −0.120665
$$964$$ 39.3432 1.26716
$$965$$ 20.1687 0.649253
$$966$$ 0 0
$$967$$ −8.22976 −0.264651 −0.132326 0.991206i $$-0.542244\pi$$
−0.132326 + 0.991206i $$0.542244\pi$$
$$968$$ 20.0738 0.645196
$$969$$ −39.6338 −1.27322
$$970$$ 5.13046 0.164729
$$971$$ −23.6326 −0.758407 −0.379204 0.925313i $$-0.623802\pi$$
−0.379204 + 0.925313i $$0.623802\pi$$
$$972$$ 16.4850 0.528758
$$973$$ 0 0
$$974$$ 6.00187 0.192312
$$975$$ 0 0
$$976$$ −11.9462 −0.382390
$$977$$ 26.6428 0.852379 0.426190 0.904634i $$-0.359856\pi$$
0.426190 + 0.904634i $$0.359856\pi$$
$$978$$ −18.3318 −0.586185
$$979$$ −5.47897 −0.175109
$$980$$ 0 0
$$981$$ 15.8600 0.506370
$$982$$ 3.47629 0.110933
$$983$$ −43.6302 −1.39159 −0.695793 0.718242i $$-0.744947\pi$$
−0.695793 + 0.718242i $$0.744947\pi$$
$$984$$ 32.1276 1.02419
$$985$$ 2.10821 0.0671732
$$986$$ −1.33187 −0.0424155
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 50.6174 1.60954
$$990$$ 11.4198 0.362945
$$991$$ −7.60816 −0.241681 −0.120841 0.992672i $$-0.538559\pi$$
−0.120841 + 0.992672i $$0.538559\pi$$
$$992$$ 27.8899 0.885506
$$993$$ −89.5581 −2.84204
$$994$$ 0 0
$$995$$ −19.1983 −0.608628
$$996$$ 64.2984 2.03737
$$997$$ 19.4356 0.615532 0.307766 0.951462i $$-0.400419\pi$$
0.307766 + 0.951462i $$0.400419\pi$$
$$998$$ 3.71523 0.117604
$$999$$ −4.38757 −0.138817
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8281.2.a.cd.1.3 6
7.6 odd 2 8281.2.a.cc.1.3 6
13.12 even 2 637.2.a.n.1.4 yes 6
39.38 odd 2 5733.2.a.br.1.3 6
91.12 odd 6 637.2.e.o.508.3 12
91.25 even 6 637.2.e.n.79.3 12
91.38 odd 6 637.2.e.o.79.3 12
91.51 even 6 637.2.e.n.508.3 12
91.90 odd 2 637.2.a.m.1.4 6
273.272 even 2 5733.2.a.bu.1.3 6

By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.4 6 91.90 odd 2
637.2.a.n.1.4 yes 6 13.12 even 2
637.2.e.n.79.3 12 91.25 even 6
637.2.e.n.508.3 12 91.51 even 6
637.2.e.o.79.3 12 91.38 odd 6
637.2.e.o.508.3 12 91.12 odd 6
5733.2.a.br.1.3 6 39.38 odd 2
5733.2.a.bu.1.3 6 273.272 even 2
8281.2.a.cc.1.3 6 7.6 odd 2
8281.2.a.cd.1.3 6 1.1 even 1 trivial